Common Core Grade 9

High School Model Curriculum for Mathematics

Summary: The High School Model Curriculum for Mathematics is not divided into individual grade standards and indicators, but instead consists of the criteria that students need to meet through high school. It is mainly made up of Algebra I, Geometry and Algebra II, with an in-depth look at each area. This new way of organizing the new standards is the "college and career ready line." The new standards are supposed to help students be better for college and the real-world when it comes to mathematics and high-level thinking.

Number and Quantity

Domain: Real Number System

Cluster: Extend the properties of exponents to rational exponents

• Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
• Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Cluster: Use properties of rational and irrational numbers

• Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Domain: Quantities

Cluster: Reason quantitatively and use units to solve problems

• Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

• Define appropriate quantities for the purpose of descriptive modeling.

• Choose a level of accuracy appropriate to limitations on measurement reporting quantities

Domain: Complex Number System

Cluster: Perform arithmetic operations with complex numbers

• Know there is a complex number i such as i2 = -1, and every complex number has the form a +bi with a and b real.
• Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
• (+) Find the conjugate of complex number; use conjugates to find moduli and quotients of complex numbers.

Cluster: Represent complex numbers and their operations on the complex plane

• Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
• Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation
• Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

Cluster: Use complex numbers in polynomial identities and equations

• Solve quadratic equations with real coefficients that have complex solutions
• (+) Extend polynomial identities to the complex numbers.
• (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

Domain: Vector and Matrix Quantities

Cluster: Represent and model with vector quantities

• Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, IvI, IIvII, v).
• Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
• Recommended to move to the next cluster

Cluster: Perform operations on vectors

• Solve problems involving velocity and other quantities that can be represented by vectors
• Add and subtract vectors.
• Multiply a vector by a scalar

Algebra

Domain: Seeing Structure in Expressions

Cluster: Interpret the structure of expressions

• Interpret expressions that represent a quantity in terms of its context. (a) Interpret parts of an expression, such as terms, factors, and coefficients. (b) Interpret complicated expressions by viewing one or more of their parts as a single entity.
• Use the structure of an expression to identify ways to rewrite it

Cluster: Write expressions in equivalent forms to solve problems

• Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression a. Factor a quadratic expression to reveal the zeros of the function it defines b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. c. Use the properties of exponents to transform expressions for exponential functions.
• Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.

Domain: Arithmetic with Polynomials and Rational Expressions

Cluster: Perform arithmetic operations on polynomials

• Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Cluster: Understand the relationship between zeros and factors of polynomials

• Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
• Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Cluster: Use polynomial identities to solve problems

• Prove polynomial identities and use them to describe numerical relationships.
• Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.

Cluster: Rewrite rational expressions

• Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
• Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

Domain: Creating Equations

Cluster: Create equations that describe numbers or relationships

• Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
• Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
• Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
• Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Domain: Reasoning with Equations and Inequalities

Cluster: Understand solving equations as a process of reasoning and explain the reasoning

• Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
• Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Cluster: Solve equations and inequalities in one variable

• Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
• Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them in a ± bi for real numbers a and b.

Cluster: Solve systems of equations

• Prove that a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
• Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
• Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3
• Represent a system of linear equations as a single matrix equation in a vector variable.
• Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3x3 or greater)

Cluster: Represent and solve equations and inequalities graphically

• Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
• Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
• Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Functions

Domain: Interpreting Functions

Cluster: Understand the concept of a function and use function notation

• Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, the f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
• Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
• Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

Cluster: Interpret functions that arise in applications in terms of the context

• For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
• Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
• Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Example:

x	y
2	5
4	10
6	15
8	20
10	25
12	30

Calculate the average rate of change within this table.

1) Pick any two points (ex. (4,10) and (10,25)
2) Use the average rate of change formula to calculate the rate of change (ex. (Y2-Y1) / (X2-X1) = (25-10) / (10-4) = 15/6 = 5/2

Cluster: Analyze functions using different representations

• Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)5, y = (0.97)5, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.

• Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Domain: Building Functions

Cluster: Build a function that models a relationship between two quantities

• Write a function that describes a relationship between two quantities. (a) Determine an explicit expression, a recursive process, or steps for calculation from a context. (b) Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential and relate these functions to the model. (c) (+) Compose functions.

• Write arithmetic and geometric sequences both recursively and with an explicit formula; use them to model situations, and translate between the two forms.

Cluster: Build new functions from existing functions

• Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
• Find inverse functions.

a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.
b. (+) Verify by composition that one function is the inverse of another.
c. (+) Read values of an inverse function from a graph or table, given that the function has an inverse.
d. (+) Produce an invertible function from a non-invertible function by restricting the domain.
5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Domain: Linear, Quadratic, and Exponential Models

Cluster: Construct and compare linear and exponential models and solve problems

• Distinguish between situations that can be modeled with linear functions and with exponential functions.

a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

• Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table.)
• Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
• For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Cluster: Interpret expressions for functions in terms of the situation they model

• Interpret the parameters in a linear or exponential function in terms of a context.

Domain: Trigonometric Functions

Cluster: Extend the domain of trigonometric functions using the unit circle

• Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
• Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
• Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4, and π/6, and use the unit circle to express the values of sine, cosines, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real number.
• Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Cluster: Model periodic phenomena with trigonometric functions

• Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
• Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
• Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of context.

Cluster: Prove and apply trigonometric identities

• Prove the Pythagorean identity sin2(θ)+cos2(θ)=1 and use it to calculate trigonometric ratios.
• Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Geometry

Congruence:

• Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment.
• Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
• Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
• ASA, SAS, SSS.
• Prove theroms about lines, angles, triangles, and parallelagrams.
• Make formal geometric constructions with a variety of tools and methods such as a compass and straightedge, string, reflective devices, paper folding, etc...

Similarities, right triangles and trigonometry:

• Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar.
• Prove theroms about triangles.
• Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
• Explain and use the relationship between the sine and cosine of complementary angles.
• Pythagorean Theorem to solve right triangles in applied problems.
• Understand and apply the Law of Sines and the Law of Cosines, and use them to solve problems and prove theroms.

Circles:

• Prove that all circles are similar.
• Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
• Construct a tangent line from a point outside a given circle to the circle.

Expressing geometric properties with equations:

• Translate between the geometric description and the equation for a conic section.
• Use coordinates to prove simple geometric theorems algebraically.

Geometric measurement and dimension:

• Explain volume formulas and use them to solve problems.
• Visualize relationships between two-dimensional and three-dimensional objects.

Modeling:

• Use geometric shapes, their measures, and their properties to describe objects.
• Apply geometric methods to solve design problems.

Example:

A triangle has sides equal to 5 cm, 10 cm and 7 cm. Find its angles (round answers to 1 decimal place).

• Let us use the figure below and set

Triangle Problem 1

• a = 10 cm , b = 7 cm and c = 5 cm.

• We now use cosine law to find the largest angle A. a 2 = b 2 + c 2- 2 b c cos(A)
• Substitute a, b and c by their values and solve for cos (A) cos (A) = [ b 2 + c 2 - a 2 ] / 2 b c cos (A) = [ 7 2 + 5 2 - 10 2 ] / (2*7*5) cos (A) = [ 7 2 + 5 2 - 10 2] / (2*7*5) = -13 / 35
• Use calculator to find angle A and round to 1 decimal place. A = arccos(-13 / 35) (approximately) = 111.8 o
• We may again use the cosine law to find angle B or the sine law. We use the sine law.a / sin (A) = b / sin(B)
• sin(B) is given by. sin (B) = (b / a) sin(A) = (7 / 10) sin (111.8 o)
• Use calculator to find B and round to 1 decimal place. B (approximately) = 40.5 o
• Use the fact that the sum of all angles in a triangle is equal to 180 o to find angle C. C (approximately) = 180 o - (40.5 o + 111.8 o) = 27.7 o

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Statistics and Probabilities

Interpreting Data:

• Represent data with plots on the real number line.
• Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages.
• Recognize possible associations and trends in the data.
• Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
• Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

Making inferences and justifying conclusions:

• Understand statistics as a process for making inferences about a population.
• Decide if a specified model is consistent with results from a given data.
• Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
• Evaluate reports based on data.

Conditional Probability and Rules:

• Understand independence and conditional probability and use them to interpret data. Use subsets, sample space, unions, intersections and complements.
• Use the rules of probability to compute probabilities of compound event.
• Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B).
• Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B).
• Use permutations and combinations to compute probabilities of more than one event.

Example:

Suppose we roll two dice and want to find the probability of rolling a sum of 6 or 8. This can be written in words as P(6 or 8) or more mathematically is P(6,8). Remember that OR (the union symbol ) means that one or the other or both events can happen. So what is the probability of getting a 6 or an 8 or both?

P(6) = 5/36 P(8) = 5/36 P(6 and 8 together) is impossible so the probability is 0.

So P(6 U 8) = 5/36 + 5/36 - 0 = 10/36 = 5/18

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